Problem A. 614. (April 2014)

A. 614. In the triangle \(\displaystyle A_1A_2A_3\), denote by \(\displaystyle k_i\) the excircle opposite to \(\displaystyle A_i\), and let \(\displaystyle P_i\) be the point of \(\displaystyle k_i\) for which the circle \(\displaystyle A_{i+1}A_{i+2}P_i\) is tangent to \(\displaystyle k_i\). (\(\displaystyle i=1,2,3\); the indices are considered modulo \(\displaystyle 3\).) Show that the line segments \(\displaystyle A_1P_1\), \(\displaystyle A_2P_2\) and \(\displaystyle A_3P_3\) are concurrent.